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2 Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different components of a right triangle Compute Per-Unit Quantities Identify the two components of Vectors

3 Right Triangles

4 Mathematics Review To be able to understand basic AC power concepts, a familiarization with the relationships between the angles and sides of a right triangle is essential A right triangle is defined as a triangle in which one of the three angles is a right angle always equal to 90 o Two of the sides which form the right triangle are designated as the adjacent and opposite sides with respect to the angle theta The third side of the right triangle is the longest side and is called the hypotenuse

6 Mathematics Review Given the lengths of two sides of a right triangle, the third side can be determined using the Pythagorean Theorem The square of the hypotenuse is equal to the sum of the squares of the remaining two sides: Hypotenuse 2 = Opposite 2 + Adjacent 2

8 Mathematics Review Once the sides are known, the next step in solving the right triangle is to determine the two unknown angles of the right triangle Three angles of any triangle always add up to 180 o In solving a right triangle, the remaining two unknown angles must add up to 90 o Basic trigonometric functions are needed to solve for the values of the unknown angles

9 Trigonometry

10 Mathematics Review The sine function is a periodic function in that it continually repeats itself

11 Mathematics Review In order to solve right triangles, it is necessary to know the value of the sine function between 0 o and 90 o Sine of either of the unknown angles of a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse SIN = Opposite side / Hypotenuse

12 Mathematics Review Cosine function is a periodic function that is identical to the sine function except that it leads the sine function by 90 o

13 Mathematics Review As an example, the cosine function at 0 o is 1 whereas the sine function does not reach the value of 1 until 90 o Cosine function of either of the unknown angles of a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse COS = Adjacent side / Hypotenuse

14 Mathematics Review The tangent function of either of the unknown angles of a right triangle is the ratio of the length of the opposite side to the length of the adjacent side TAN = Opposite side / Adjacent side

15 Mathematics Review Example: Given: Side H = 5, Side A = 4 Find: Side O, Angle and Angle Hypotenuse Side H 5 α Opposite Side Side O θ 90 o 4 Adjacent Side Side A

26 Mathematics Review Ratios play an important part in estimating power system performance A ratio is defined as a relationship between two numbers as a fraction Generally, ratios are used when the relationship of two pairs of values is the same, and one of two similarly related values is known Ratios only provide exact answers in linear systems where the relationship between two variables in the system is the same regardless of the magnitude of the two variables

27 Question 4 Assume that the loss of a 1000 MW generating unit will typically result in a 0.2 Hz dip in system frequency. Estimate the frequency dip for the loss of an 800 MW generating unit.

31 Mathematics Review Quantities on the power system are often specified as a percentage or a per-unit of their base or nominal value Per-unit values makes it easier to see where a system value is in respect to its base value and also how it compares between different parts of the system with different base values Per-unit values also allow for a dispatcher to view the system and quickly obtain a feel for the voltage profile

35 Vectors A vector is alternative way to represent a sinusoidal function with amplitude, and phase information A vectors length represents magnitude A vectors direction represents the phase angle Example: 10 miles east N W 10 Miles E S

36 Vectors Vectors are a means of expressing both magnitude and direction Horizontal line to the right is positive; horizontal line to the left is negative Vertical line going up is positive; vertical line going down is negative Arrowhead on the end away from the point of origin indicates the direction of the vector and is called the displacement vector Vectors can go in any direction in space

37 Vectors The difference between a scalar quantity and a vector: a) A scalar quantity is one that can be described with a single number, including any units, giving its size or magnitude b) A vector quantity is one that deals inherently with both magnitude and direction

38 Conceptual Question 6 There are places where the temperature is +20 o C at one time of the year and -20 o C at another time. Do the plus and minus signs that signify positive and negative temperatures imply that temperature is a vector quantity?

39 Question 7 Which of the following statements, if any, involves a vector? a) I walked two miles along the beach b) I walked two miles due north along the beach c) A ball fell off a cliff and hit the water traveling at 17 miles per hour d) A ball fell off a cliff and traveled straight down 200 feet e) My bank account shows a negative balance of -25 dollars

40 Vectors When adding vectors, the process must take into account both the magnitude and direction of the vectors Vectors are usually written in bold with an arrow over the top of the letter, ( A ) When adding two vectors, there is always a resultant vector, R, and the addition is written as follows: R = A + B

41 Vectors Example: Adding vectors in the same direction Vector A has a length of 2 and a direction of 90 o Vector B has a length of 3 and a direction of 90 o 90 o B R 180 o A 0 o 270 o

42 Vectors Example: Adding vectors in the opposite direction Vector A has a length of 2 and a direction of 90 o Vector B has a length of 3 and a direction of o A 180 o 0 o R B 270 o

43 Question 8 Two vectors, A and B, are added to give a resultant vector, R. The magnitudes are 3 and 8 meters, respectively, but the vectors can have any orientation. What is the maximum and minimum possible values for the magnitude of R? 90 o Maximum: 3+8=11 A 180 o 0 o Minimum: (-3)+(-8)=(-11) B 270 o

44 Vectors Subtraction of one vector from another is carried out in a way that depends on the following: When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed Vector subtraction is carried out exactly like vector addition except that one of the vectors added is multiplied by the scalar factor of -1

45 Vectors C B C B A A C = A + B A = C + (-B)

46 Vectors If the magnitude and direction of a vector is known, it is possible to find the components of the vector The process is called resolving the vector into its components If the vector components are perpendicular and form a right triangle, the process can be carried out with the aid of trigonometry

47 Vectors To calculate the sum of two or more vectors using their components (x and y) in the vertical and horizontal directions, trigonometry is used Opposite Side (X) θ Adjacent Side (Y)

56 Vectors Polar notation expresses a vector in terms of both a magnitude and a direction such as: where: M M is the magnitude of the vector o is the direction in degrees Example: Vector with a magnitude of 10 and a direction of -40 degrees o

60 Summary Discussed the different components of a Right Triangles Reviewed the basics of Trigonometry Computed different Per-Unit Quantities Characterized the two components of Vectors

61 Questions?

62 Disclaimer: PJM has made all efforts possible to accurately document all information in this presentation. The information seen here does not supersede the PJM Operating Agreement or the PJM Tariff both of which can be found by accessing: For additional detailed information on any of the topics discussed, please refer to the appropriate PJM manual which can be found by accessing:

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